Philosophy

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The similarity metric

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A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new "normalized information distance," based on the noncomputable notion of
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  The UMAP Journal Publisher COMAP, Inc.  Vol. 3   , No.   Executive Publisher Solomon A. Garfunkel ILAP Editor Chris ArneyDept. of Math’l SciencesU.S. Military AcademyWest Point, NY 10996 david.arney@usma.edu On Jargon Editor Yves NievergeltDept. of MathematicsEastern Washington Univ.Cheney, WA 99004 ynievergelt@ewu.edu Reviews Editor  James M. CargalMathematics Dept.Troy University—Montgomery Campus231 Montgomery St.Montgomery, AL 36104 jmcargal@sprintmail.com Chief Operating Of  cer Laurie W. Arag´on Production Manager George Ward Copy Editor  Julia Collins Distribution  John Tomicek Editor Paul J. CampbellBeloit College700 College St.Beloit, WI 53511–5595 campbell@beloit.edu Associate Editors Don AdolphsonAaron ArcherChris ArneyRon BarnesArthur BenjaminRobert Bosch James M. CargalMurray K. ClaytonLisette De Pillis James P. FinkSolomon A. GarfunkelWilliam B. GearhartWilliam C. GiauqueRichard Haberman Jon JacobsenWalter MeyerYves NievergeltMichael O’LearyCatherine A. Roberts John S. RobertsonPhilip D. Straf   n J.T. SutcliffeBrigham Young Univ.AT&T Shannon Res. Lab.U.S. Military AcademyU. of Houston—DowntnHarvey Mudd CollegeOberlin CollegeTroy U.— MontgomeryU. of Wisc.—MadisonHarvey Mudd CollegeGettysburg CollegeCOMAP, Inc.Calif. State U., FullertonBrigham Young Univ.Southern Methodist U.Harvey Mudd CollegeAdelphi UniversityEastern Washington U.Towson UniversityCollege of the Holy CrossGeorgia Military CollegeBeloit CollegeSt. Mark’s School, Dallas  Subscription Rates for 2012 Calendar Year: Volume 33 Institutional Web Membership (Web Only) Institutional Web Memberships do not provide print materials. Web memberships allowmembers to search our online catalog, download COMAP print materials, and reproduce them for classroom use. (Domestic) #3030 $467 (Outside U.S.) #3030 $467 Institutional Membership (Print Only) Institutional Memberships receive print copies of The UMAP Journal quarterly, our annual CD collection UMAP Modules, Tools for Teaching, and our organizational newslet-ter Consortium. (Domestic) #3040 $312 (Outside U.S.) #3041 $351 Institutional Plus Membership (Print Plus Web) Institutional Plus Memberships receive print copies of the quarterly issues of The UMAP Journal, our annual CD collection UMAP Modules, Tools for Teaching, our organizationalnewsletter Consortium, and online membership that allows members to search our online catalog, download COMAP print materials, and reproduce them for classroom use. (Domestic) #3070 $615 (Outside U.S.) #3071 $659 For individual membership options visit www.comap.com for more information.To order, send a check or money order to COMAP, or call toll-free 1-800-77-COMAP (1-800-772-6627). The UMAP Journal is published quarterly by the Consortium for Mathematics and Its Applications (COMAP), Inc., Suite 3B, 175 Middlesex Tpke.,Bedford, MA, 01730, in coop-eration with the American Mathematical Association of Two-Year Colleges (AMATYC), the Mathematical Association of America (MAA), the National Council of Teachers of Mathematics (NCTM), the American Statistical Association (ASA), the Society for Industrial and Applied Mathematics (SIAM), and The Institute for Operations Re-search and the Management Sciences (INFORMS). The Journal acquaints readers with awide variety of professional applications of the mathematical sciences and provides a forumfor the discussion of new directions in mathematical education (ISSN 0197-3622). Periodical rate postage paid at Bedford, MA and at additional mailing offices. Send address changes to: info@comap.com COMAP, Inc., Suite 3B, 175 Middlesex Tpke., Bedford, MA, 01730 © Copyright 2012 by COMAP, Inc. All rights reserved. Mathematical Contest in Modeling (MCM) ® , High School Mathematical Contest inModeling (HiMCM) ® , and Interdisciplinary Contest in Modeling(ICM) ® are registered trade marks of COMAP, Inc.  Vol. 3 3  , No . 3   012Tableof ContentsGuestEditorial NetworkScience: What’s MathGotto DowithIt? Chris Arney ..............................................................................185 Editor’s Note AboutThisIssue ...........................................................................192 MCMModelingForum Resultsof the2012MathematicalContestin Modeling William P. Fox ...........................................................................193 A CloseLookat Leaves Bo Zhang, Yi Zhang, and TianKun Lu .........................................205  Judges’Commentary: TheOutstandingLeafProblemPapers Peter Olsen................................................................................223 ComputingAlongtheBigLongRiver Chip Jackson, Lucas Bourne, and Travis Peters ............................231  Judges’Commentary: TheOutstandingRiverProblemPapers Marie Vanisko ...........................................................................247 Author’s Commentary: TheOutstandingRiverProblemPapers Catherine A. Roberts..................................................................253  Judges’Commentary: TheGiordanoAwardfortheRiverProblem Marie Vanisko and Richard D. West............................................259 ICMModelingForum Resultsof the2012InterdisciplinaryContestin Modeling Chris Arney ..............................................................................263 FindingConspiratorsin theNetworkviaMachineLearning Fangjian Guo, Jiang Su, and Jian Gao..........................................275  Judges’Commentary: ModelingforCrimeBusting Chris Arney and Kathryn Coronges............................................293 Reviews ...............................................................................305  Guest Editorial  185 Guest Editorial Network Science:What’s Math Got to Do with It?  1 Chris Arney Dept. of Mathematical SciencesU.S. Military AcademyWest Point, NY 10996 david.arney@usma.edu Introduction Thisyear’sICM R  probleminvolvednetworkscience,ormoreprecisely,acomponentofnetworkscience—socialnetworkanalysis. Mypost-contestre fl ectionshaveledmetobelieveitistimeforthemathematicscommunityto engage in this emerging subject to build a rigorous mathematical foun-dation for this important science and to join in performing mathematicalmodeling and interdisciplinary problem solving.Some people call network science a “new” emerging discipline, yet, aswe know, mathematicians have been developing graph (network) theoryfor centuries, and scientists and engineers have been modeling networksfor decades. What is new is that the traditional techniques have been re-placed by an entirely new arsenal of mathematics, science, and modelingassociated with networks.Otherscallnetworksciencethe“new”operationsresearchinthatitcon-nects quantitative concepts and elements from several disciplines such asmathematics, computer science, and information science with the qualita-tive models from sociology and other social sciences. By its very nature,networkscienceisinterdisciplinaryandinvolvesemergingareasofsciencesuch as complex adaptive systems, cooperative game theory, agent-basedmodeling, data analytics, and social network analysis. 1 With both appreciation and apologies to Tina Turner and her emotional song “What’s LoveGotToDoWithIt”andfulltongue-in-cheekrealizationthatunlike“love,”mathematicsiscertainlynot a second-hand emotion. TheUMAPJournal 33(3)(2012)185–191. c  Copyright2012byCOMAP,Inc. Allrightsreserved.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for pro fi t or commercialadvantage and that copies bear this notice. Abstracting with credit is permitted, but copyrightsfor components of this work owned by others than COMAP must be honored. To copy otherwise,to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP.  186  The UMAP Journal 33.3 (2012) From another, perhaps simpler, perspective, you could merely call net-work science a form of applied mathematics—applieddynamic graph the-ory with additional data elements and attributes. Or perhaps from a mod-eling perspective, it is simply modeling with a highly structured, entity-linked complex adaptive system framework.I do not pretend to know exactly what network science is, or where it fi ts in with today’s scienti fi c world, or what it will become. However, I believe that much of the strength of network modeling is in its ability toembrace the complexity of the real world. For me, that makes networkscience an important and empowering form of interdisciplinary modelingand problem solving—worthy of ICM problems and much more.In particular, I hope that the mathematics community does not ignoreit. Network science needs the engagement of the mathematics communityto produce its underlying framework and to invent new mathematics andcomputational techniques for analysis of its complex structures, develop-ment of its synergistic processes, and organizing of its overwhelmingdata.Likewise, mathematics needs network science to establish the relevance of mathematics in the modern information-based world. As the ICM teamsdiscovered, network science is exciting, relevant, enjoyable, and modern—elements that mathematicscurrently desperatelyneeds to bolster its futureplace in society. Mathematical Elements What are the mathematical elements of network science? One way tode fi ne a network is to establish its •  components (nodes, links, data, processes); •  properties (dynamic, functional, layered); and •  applications(logistics, fl ow,transportation,Internet,metabolicnetworks,social networks, organizational networks—perhaps there are just toomany categories to list!).Another way is to use the concept of a mathematical graph (the nodal-link structure) with its nontrivial topological features and then classify thevarious types of graphs that occur (random, scale free, small world, scalerich) and the data (often heavy-tailed)that need to be mined and analyzed.Afoundationalresearchmanagementreportonnetworkscienceoffereda layered approach of network roles—physical, communicative, informa-tional, biological and social/cognitive—that connect together to producetheoverallweb-likenetworkframework[NationalResearchCouncil2006].Nomatterwhatde fi nitionortheoreticalframeworkisused,networksci-enceisinherentlyandessentiallymathematicalatitscore;thereisplentyforappliedmathematicianstodo. Mostnetworksaresuf  fi cientlycomplexthat
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